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My question, is primarily, based on this question, in which the accepted answer asserts that $\vec B \, $ has two definitions,

  1. As magnetic field$^*$, using Lorentz force $\vec F=q \vec v \times \vec B $
  2. As magnetic flux density, from $\Phi=\int\vec B.d\vec S$

My question is how do these two definitions, result in the same vector field?


$*$ I'm not able to define $\vec B$ as I could define $\vec E$, as force per unit charge. Is there one such definition for $\vec B$, or was it introduced, so that we could formulate our observations easily with such a vector field?

In laws of induction, we use B as flux density and while finding the Lorentz force, we use it as a vector field whose magnitude is given by Biot-Savart law. How are these two equal?

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  • $\begingroup$ I don't think the answer you refer to really does claim that these are both definitions. The relation involving $\Phi$ can't define $B$ unless you already have a definition of $\Phi$. $\endgroup$ – Ben Crowell Sep 14 at 13:51
  • $\begingroup$ @BenCrowell Then I really don't understand that answer and my question is the same as that. In laws of induction, we use B as flux density and while finding the Lorentz force, we use it as a vector field whose magnitude is given by Biot-Savart law. How are these two equal? $\endgroup$ – Aravindh Vasu Sep 14 at 13:54

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